Klir G.J. Uncertainity and Information. Foundations of Generalized Information Theory

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under the following speciﬁcations:

(a) x = 65% and T = True;

(b) x = 65% and T = False;

(d) x = 50% and T = False;

(e) x = 76% and T = Fairly true;

(f) x = 76% and T = Very false.

312 7. FUZZY SET THEORY

Table 7.2. Matrix Representations of Fuzzy Relations Employed in Exercises 7.25–7.27

1.0 0.0 0.7

= x

0.3 0.2 0.0

0.0 0.5 1.0

1.0 0.9 0.8 0.7

0.6 1.0 0.5 0.4

0.3 0.2 1.0 0.1

0.2 0.3 0.4 1.0

0.9 1.0 1.0 1.0 0.9

0.7 0.8 0.9 0.7 0.6

0.6 0.5 0.8 0.6 0.5

0.3 0.4 0.7 0.3 0.4

0.2 0.5 0.6 0.5 0.2

0.1 0.3 0.5 0.3 0.1

1.0 0.0 0.7 0.5

= y

0.0 1.0 0.0 1.0

0.7 0.0 1.0 0.8

0.0 0.8 0.6 0.0 0.4 0.2

0.0 0.0 0.9 0.9 0.7 0.7

1.0 1.0 0.0 0.5 0.0 0.0

1.0 1.0 0.5 0.0 1.0 1.0

1.0 0.9 0.9 0.8 0.6 0.4 0.2

1.0 0.9 1.0 0.8 0.7 0.7 0.5

1.0 1.0 0.8 0.9 0.0 0.0 0.0

0.0 0.0 0.0 0.7 0.8 0.0 0.0

0.0 0.0 0.0 0.0 0.6 0.7 0.0

0.0 0.0 0.0 0.5 0.5 0.5 0.6

7.29. Repeat Exercise 7.28 for the property of medium humidity, which

is expressed by the trapezoidal-shaped fuzzy interval M =·20, 40, 60,

80Ò.

7.30. The probabilities, p(r), of daily receipts, r, of a shop (rounded to

the nearest hundred dollars) that have been obtained from statistical

data collected over many years are shown in Table 7.1b. Consider three

fuzzy events: low, medium, and high receipts and assume that they are

represented by trapezoidal-shaped fuzzy intervals L =·18, 18, 21, 23Ò,

M =·21, 23, 27, 29Ò, and H =·27, 29, 32, 32Ò, respectively. Determine the

degrees of truth of the following propositions:

(a) The daily receipts are low;

(b) The daily receipts are medium;

7.31. Modify the granulation deﬁned in Figure 7.10b by replacing the trian-

gular-shaped granules with:

(a) Trapezoidal-shaped granules, with small plateaus around the ideal

values;

(b) Granules of the shape illustrated by membership function C in

Figure 7.4 and deﬁned in Exercise 7.4;

Figure 7.4 and deﬁned in Example 7.3.

7.32. Show that the defuzziﬁed value d(A) obtained by the centroid defuzzi-

ﬁcation method may be interpreted as the expected value of x based on

7.33. Show that the defuzziﬁed value d(A) deﬁned by Eq. (7.41) is the value

of x for which the area under the graph of membership function A is

divided into two equal areas.

7.34. Using the centroid defuzziﬁcation method defuzzify the following fuzzy

sets:

(a) Fuzzy number A(x) = max{0, 4x - 4 - (x - 2)

};

(b) Fuzzy interval L in Figure 7.7a;

Figure 7.5;

(d) Fuzzy set A that is equal to the possibility proﬁle r in Figure 6.1 (i.e.,

A(x) = r(x) for all x Œ ⺞

);

(e) Fuzzy interval A deﬁned by the following a-cut representation:

aaa

aa a

A =

()

[]

(

]

[]

(

]

15 1754 05 0 05

254 05 051

..,. ,.

., . ., .

when

EXERCISES 313

7.35. Repeat Exercise 7.34 for some other defuzziﬁcation methods deﬁned by

Eqs. (7.43) and (7.44). Choose at least one value of l < 1 and one value

of l > 1.

7.36. Every rectangle may be considered to some degree a member of a fuzzy

set of squares. Using common sense, deﬁne a possible membership func-

tion of such a fuzzy set.

314 7. FUZZY SET THEORY

FUZZIFICATION OF

UNCERTAINTY THEORIES

315

The limit of language is the limit of the world.

—Stephen A. Tyler

8.1. ASPECTS OF FUZZIFICATION

Perhaps the best description of the nature of fuzziﬁcation is contained in the

well-known classical paper by Goguen [1967]: “Fuzziﬁcation is a process of

imparting a fuzzy structure to a deﬁnition (concept), a theorem, or even a

whole theory.” The strength of this deceivingly simple description is its sweep-

ing generality. The single sentence captures the essence of a wide variety of

issues that must be addressed when mathematical concepts, properties, or the-

ories based on classical sets are generalized to their counterparts based on

fuzzy sets of some type.

One method of fuzzifying various properties (concepts, operations,

theorems) of classical set theory is to use the a-cut (or strong a-cut) repre-

sentations of fuzzy sets. Since a-cuts (as well as strong a-cuts) are classical sets,

every property of classical set theory applies to them. A property of classical

sets is fuzziﬁed (that is, it becomes a property of fuzzy sets) via the a-cut rep-

resentation by requiring that it holds (in the classical sense) in all a-cuts of

the fuzzy sets involved. Any property of fuzzy sets of some speciﬁc type that

is derived from a property of classical sets in this way is called a cutworthy

property.

For standard fuzzy sets, there are many properties that are cutworthy. One

such property is convexity of fuzzy sets. A fuzzy set is said to be convex if and

Uncertainty and Information: Foundations of Generalized Information Theory, by George J. Klir

only if all its a-cuts are convex sets in the classical sense. An important class

of special convex fuzzy sets consists of fuzzy intervals. A fuzzy set on ⺢ is said

to be a fuzzy interval if and only if all its a-cuts are closed intervals of real

numbers (i.e., classical convex subsets of ⺢). Arithmetic operations on fuzzy

intervals are also cutworthy.At each a-cut, they follow the rules of either stan-

dard or constrained arithmetic on closed intervals of real numbers.

It is signiﬁcant that the standard operations of intersection and union of

fuzzy sets (min and max operations) are cutworthy. As a consequence, other

operations that are based solely on them are cutworthy as well. They include,

for example, the max–min composition of binary fuzzy relations as well as the

relational join deﬁned by the min operation. The concept of cylindric closure

of fuzzy relations is also cutworthy when the intersection of the cylindric

extensions of the given fuzzy relations is deﬁned by the min operation.

Important examples of cutworthy properties are some properties of binary

fuzzy relations on X

, such as equivalence, compatibility, or partial ordering.

Classical equivalence relations, for example, are deﬁned by three properties:

(i) reﬂexivity; (ii) symmetry; and (iii) transitivity. Let these properties be

deﬁned for fuzzy relations, R, as follows:

(i) R is reﬂexive iff R(x, x) = 1 for all x ŒX.

(ii) R is symmetric iff R(x, y) = R(y, x) for all x,y ŒX.

(iii) R is transitive (or, more speciﬁcally, max–min transitive) iff

(8.1)

for all pairs ·x,zÒŒX

Then, any binary fuzzy relation on X

that possesses these properties is a fuzzy

equivalence relation in the cutworthy sense. That is, all a-cuts of any binary

fuzzy relation that possesses these properties are classical equivalence rela-

tions. This follows from the fact that each of the three properties is cutworthy.

In the case of reﬂexivity and symmetry, it is obvious. In the case of transitiv-

ity, it is cutworthy since it is deﬁned in terms of the standard operations of

intersection and union, which are cutworthy.

EXAMPLE 8.1. Consider the binary fuzzy relation R on X

, where X =

|i Œ ⺞

}, which is deﬁned by the matrix

xxxxxxx

1234567

R =

000

Rxz Rx y Ryz

,maxmin,,,

()

≥

()(){}

316 8. FUZZIFICATION OF UNCERTAINTY THEORIES

It is easy to see that this relation is reﬂexive and symmetric. It is also max–min

transitive, but that is more difﬁcult to verify. One convenient way to verify it

is to calculate (R

R) » R where

denotes the max–min composition and »

denotes the standard union operation. Then, R is transitive if and only if

The given relation satisﬁes this equation. Hence, it is max–min transitive and,

due to its reﬂexivity and symmetry, it is a fuzzy equivalence relation. This can

be veriﬁed by examining all its a-cuts.

The level set of the given relation R is L

= {0, 0.4, 0.5, 0.8, 0.9, 1}. There-

fore, R represents six classical equivalence relations, one for each a ŒL

. Each

of these equivalence relations,

R, partitions the set X in some particular way,

p(X). Since

a ¢

R 債

R when a¢≥a, clearly

The six partitions of the given relation are shown in the form of a partition

tree in Figure 8.1. The partitions become increasingly more reﬁned when

values of a in L

increase.

Other cutworthy types of binary fuzzy relations can be deﬁned in a similar

way. Examples are fuzzy compatibility relations (reﬂexive and symmetric) and

fuzzy partial orderings (reﬂexive, antisymmetric, and transitive). For fuzzy

partial orderings, the property of fuzzy antisymmetry is deﬁned as follows: for

all x, y Œ X, if R(x, y) > 0 and R(y, x) > 0, then x = y. This, clearly, is a cut-

worthy property.

It is important to realize that there are many fuzzy-set generalizations of

properties of classical sets that are not cutworthy. A fuzzy-set generalization

of some classical property is required to reduce to its classical counterpart

when membership grades are restricted to 0 and 1, but it is not required to be

cutworthy.There often are multiple generalizations of a classical property, but

only one or, in some cases, none of them is cutworthy. Examples of fuzzy-set

generalizations that are not cutworthy are all operations of intersection and

union of fuzzy sets (t-norms and t-conorms) except the standard ones (min

and max). Even more interesting examples are operations of complementa-

tion of fuzzy sets, none of which is cutworthy, even though all of them are, by

deﬁnition, generalizations of the classical complementation.

Another way of connecting classical set theory and fuzzy set theory is to

fuzzify functions. Given a function

where X and Y are crisp sets, we say that the function is fuzziﬁed when it is

extended to act on fuzzy sets deﬁned on X and Y. That is, the fuzziﬁed func-

tion maps, in general, fuzzy sets deﬁned on X to fuzzy sets deﬁned on Y.

Formally, the fuzziﬁed function, F, has a form

fX Y:,Æ

()

¢≥

pp aaXXwhen .

RRR R=

()

»o .

8.1. ASPECTS OF FUZZIFICATION 317

where F(X ) and F(Y) denote the fuzzy power sets (sets of all fuzzy subsets)

of X and Y, respectively. To qualify as a fuzziﬁed version of f, function F must

conform to f within the extended domain F(X) and F(Y). This is guaranteed

when a principle is employed that is called an extension principle. According

to this principle

is determined for any given fuzzy set A ŒF(X) and all y ŒY via the formula

(8.2)

Ax x X f x y

()

{}

()

π∆

sup ,

when

otherwise

BFA=

()

FX Y:,FF

()

318 8. FUZZIFICATION OF UNCERTAINTY THEORIES

= 0.0

= 0.4

= 0.5

= 0.8

= 0.9

= 1.0

Figure 8.1. Partition tree of the fuzzy equivalence relation in Example 8.1.

The inverse function

of F is deﬁned, according to the extension principle, for any given B ŒF(Y)

and all x ŒX, by the formula

(8.3)

where y = f(x). Clearly,

for all A ŒF(X ), where the equality is obtained when f is a one-to-one

function.

The use of the extension principle is illustrated in Figure 8.2, which shows

how fuzzy set A is mapped to fuzzy set B via function F that is consistent with

the given function f. That is, B = F(A). For example, since

we have

Bb Aa Aa Aa

()

() () (){}

max , ,

123

bfa fa fa=

()

123

FFA A

()

[]



FBx By

()

[]

()

FY X

()

: FF

8.1. ASPECTS OF FUZZIFICATION 319

(y)

A(a )

B(b)

A(x)

A(a )

A(x)

Figure 8.2. Illustration of the extension principle for fuzzy sets.

by Eq. (8.2). Conversely,

by Eq. (8.3).

The introduced extension principle, by which functions are fuzziﬁed, is basi-

cally described by Eqs. (8.2) and (8.3). These equations are direct generaliza-

tions of similar equations describing the extension principle of classical set

theory. In the latter, symbols A and B denote characteristic functions of crisp

sets.

To fuzzify a function of n variables of the form

the formula in Eq. (8.2) has to be replaced with the more general formula

(8.4)

Similarly, Eq. (8.3) has to be replaced with

(8.5)

Equation (8.4) can be further generalized by replacing the min operator with

a t-norm.

EXAMPLE 8.2. Consider the function y = , where x Œ [0, 5], and the tri-

angular fuzzy number

which represents an approximate assessment of value x. Using the extension

principle, the value of y is approximately assessed by the fuzzy number

Fuzziﬁcation also can be attained via the mathematical theory of categories.

In this way, a fuzzy structure is imparted into various categories of mathe-

matical objects by fuzzifying morphisms through which the categories are

By x

xx=

()

-Œ

[

)

-Œ

[]

21 051

32 115

when

otherwise.

()

-Œ

[

)

-Œ

[]

21 051

32 115

when

otherwise,

FBxx x By

()

[]

◊◊◊

()

,, , .

Ax x X i fx x x y f y

ii i i n n

()

ŒŒ

◊◊◊

()

{}()

π∆

sup min , , , , ,

⺞

when

otherwise.

fX X X Y

...

¥¥¥Æ

FBa FBa FBa Bb

-- -

()

[]

()

[]

()

[]

()

320 8. FUZZIFICATION OF UNCERTAINTY THEORIES

deﬁned. This is a powerful approach to fuzziﬁcation, which has played an

important role in fuzzifying many areas of mathematics, such as topology,

analysis, various algebraic theories, graphs, hypergraphs, geometry, and ﬁnite-

state automata. Because the reader of this book is not expected to have sufﬁ-

cient background in category theory, this approach to fuzziﬁcation is not

employed in this chapter.

There are usually multiple ways in which a given classical mathematical

structure can be fuzziﬁed. These ways of fuzziﬁcation are distinguished from

one another by choosing which components of the structure are going to be

fuzziﬁed and how. These choices have to be made in each given application

context on pragmatic grounds.

8.2. MEASURES OF FUZZINESS

The pragmatic value of fuzzy logic in the broad sense consists of its capabil-

ity to represent and deal with vague linguistic expressions. Such expressions

are typical of natural language. A linguistic expression is vague when its

meaning is not ﬁxed by sharp boundaries. Such a linguistic expression is always

associated with uncertainty regarding its applicability. However, this type of

uncertainty does not result from any information deﬁciency, but from the lack

of linguistic precision. It is a linguistic uncertainty rather than information-

based uncertainty. Consider, for example, a fuzzy set that was constructed in a

given application context to represent the linguistic term “high temperature.”

Assume now that a particular measurement of the temperature was taken (say

92°F). This measurement belongs to the fuzzy set with a particular member-

ship degree (say, 0.7). Clearly, this degree does not express any lack of infor-

mation (the actual value of the temperature is known—it is 92°F), but rather

the degree of compatibility of the known value with the imprecise (vague)

linguistic term.

Since this linguistic uncertainty is represented by fuzzy sets, it is usually

called fuzziness.Each fuzzy set is clearly associated with some amount of fuzzi-

ness, and it is desirable to be able to measure this amount in some meaning-

ful way. Although the amount of fuzziness is not connected in any way to the

quantiﬁcation of information, it is an important trait of information represen-

tation. Clearly, our ability to measure it allows us to characterize information

representation more completely, and thus enriches our methodological capa-

bilities for dealing with information.

In general, a measure of fuzziness for some type of fuzzy sets is a functional

where F(X ) denotes the set of all fuzzy subsets of X of a given type. Thus far,

however, the issue of measuring fuzziness has been addressed only for stan-

dard fuzzy sets. This restriction is thus followed in this section as well.

fX:,F

()

⺢

8.2. MEASURES OF FUZZINESS 321